When you're presented with a weird problem in a computing contest, it's odds-on it'll be dynamic programming (the Bellman kind, not the Ruby kind.)
Have a look at this tutorial.
The basic idea of dynamic programming is to build up a "big" answer by answering small subproblems. My first intuitive thought is to think about longer and longer necklaces, starting with, like, one bead, but I'm honestly not particularly good at dynamic programming problems. (I've also noticed that the most common place to run into DP problems in the real world is in computing olympiads and problem sets in computer science classes.)
